Inequalities for the Gamma Function and estimates for the volume of sections of $B_p^n$

Sections of functions and Sobolev type inequalities

We study functions of two variables whose sections by the lines parallel to the coordinate axis satisfy Lipschitz condition of the order $0<\a\le 1.$ We prove that if for a function $f$ the $\operatorname{Lip} \a-$ norms of these sections belong to the Lorentz space $L^{p,1}(\R) \,(p=1/\a),$ then $f$ can be modified on a set of measure zero so as to become bounded and uniformly continuous on $\R^2.$ For $\a=1$ this gives an extension of Sobolev's theorem on continuity of functions of the space $W_1^{2,2}(\R^2)$. We show that the exterior $L^{p,1}-$ norm cannot be replaced by a weaker Lorentz n…

Inequalities for sections and projections of convex bodies

This article belongs to the area of geometric tomography, which is the study of geometric properties of solids based on data about their sections and projections. We describe a new direction in geometric tomography where different volumetric results are considered in a more general setting, with volume replaced by an arbitrary measure. Surprisingly, such a general approach works for a number of volumetric results. In particular, we discuss the Busemann-Petty problem on sections of convex bodies for arbitrary measures and the slicing problem for arbitrary measures. We present generalizations of…

投影函数的拉东变换与亚历山德罗夫-芬切尔不等式

Radon Transforms of Projection Functions and Aleksandrov-Fenchel Inequalities

本文针对平均截面体的对数凹性问题，建立新的亚历山德罗夫-芬切尔不等式，证明平均截面体的所有内在体积均为对数凹的。

Slicing inequalities for measures of convex bodies

We consider a generalization of the hyperplane problem to arbitrary measures in place of volume and to sections of lower dimensions. We prove this generalization for unconditional convex bodies and for duals of bodies with bounded volume ratio. We also prove it for arbitrary symmetric convex bodies under the condition that the dimension of sections is less than $λn$ for some $λ\in (0,1).$ The constant depends only on $λ.$ Finally, we show that the behavior of the minimal sections for some measures may be different from the case of volume.

Progress on Hardy-type Inequalities

This paper surveys some of our recent progress on Hardy-type inequa\-lities which consist of a well-known topic in Harmonic Analysis. In the first section, we recall the original probabilistic motivation dealing with the stability speed in terms of the $L^2$-theory. A crucial application of a result by Fukushima and Uemura (2003) is included. In the second section, the non-linear case (a general Hardy-type inequality) is handled with a direct and analytic proof. In the last section, it is illustrated that the basic estimates presented in the first two sections can still be improved considerabl…

When does $e^{-\lvert τ\rvert }$ maximize Fourier extension for a conic section?

In the past decade, much effort has gone into understanding maximizers for Fourier restriction and extension inequalities. Nearly all of the cases in which maximizers for inequalities involving the restriction or extension operator have been successfully identified can be seen as partial answers to the question in the title. In this survey, we focus on recent developments in sharp restriction theory relevant to this question. We present results in the algebraic case for spherical and hyperbolic extension inequalities. We also discuss the use of the Penrose transform leading to some negative an…